Accurately Counting Singular Values of Bidiagonal Matrices

We have developed algorithms to count singular values of a bidiagonalmatrix which are greater than a speci ed value This requires the transformation of the singular value problem to an equivalent symmetric eigenvalue problem The counting of sin gular values is paramount in the design of bisection and multisection type algorithms for computing singular values on serial and parallel machines The algorithms are based on the eigenvalues of BB BB and the n n zero diagonal tridiagonal matrix which is permutationally equivalent to the Jordan Wielandt form B B t where B is an n n bidiagonal matrix The two product matrices which do not have to be formed explicitly lead to the progressive and station ary qd algorithms of Rutishauser The algorithm based on the zero diagonal matrix which we have named the Golub Kahan form may be considered as a combination of both the progressive and stationary qd algorithms We study important properties such as the backward error analysis the mono tonicity of the inertia count and the scaling of data which guarantee the accuracy and the integrity of these algorithms For high relative accuracy of tiny singular val ues the algorithm based on the Golub Kahan form is the best choice However if such accuracy is not required or requested the di erential progressive and di erential stationary qd algorithms with certain modi cations are adequate and more e cient